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A Splitter for Graphs with No Petersen Family Minor John Maharry*

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A Splitter for Graphs with No Petersen Family Minor John Maharry* Journal of Combinatorial Theory, Series B TB1800 Journal of Combinatorial Theory, Series B 72, 136139 (1998) Article No. TB971800 A Splitter for Graphs with No Petersen Family Minor John Maharry* Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174 E-mail: maharryÄmath.ohio-state.edu Received December 2, 1996 The Petersen family consists of the seven graphs that can be obtained from the Petersen Graph by Y2- and 2Y-exchanges. A splitter for a family of graphs is a maximal 3-connected graph in the family. In this paper, a previously studied graph, Q13, 3 , is shown to be a splitter for the set of all graphs with no Petersen family minor. Moreover, Q13, 3 is a splitter for the family of graphs with no K6 -minor, as well as for the family of graphs with no Petersen minor. 1998 Academic Press View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector In this paper all graphs are assumed to be finite and simple. A graph H is called a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. This is denoted Hm G. A simple graph with at least 4 vertices is 3-connected if it remains connected after the deletion of any two vertices. Let y be a vertex of a graph G such that y has exactly three distinct neighbors [a, b, c]. Let H be the graph obtained from G by deleting y and adding a triangle on the vertices, [a, b, c]. The graph H is said to be obtained from G by a Y2-exchange at y. The inverse operation is called a 2Y-exchange at (abc). The Petersen family is the set of seven graphs that can be obtained from the Petersen graph by repeated Y2- and 2Y-exchanges. (See Fig. 1.) A graph G is said to be linkless if there exists an embedding of G into 3-space so that it contains no non-trivial link (in the sense of knot theory). A graph is said to be flat if it admits an embedding into 3-space such that every circuit of the graph bounds an open disk disjoint from the graph. Robertson et al. [3] showed that the family of flat graphs is equal to the family of linkless graphs which, in turn, is equal to the family of graphs with no minor isomorphic to a graph in the Petersen family. Suppose that a graph H is obtained from a graph G by contracting the edge with endvertices x and y (x{ y) to a new vertex w. Then G is said to be obtained from H by splitting the vertex w into the vertices x and y. A graph G is called a wheel if there exists a vertex v such that G"v is a circuit and v is adjacent to every vertex of the circuit. * The author's research was supported in part through the Office of Naval Research by Grant N00014-92-J-1965. 136 0095-8956Â98 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved. File: DISTIL 180001 . By:CV . Date:28:01:98 . Time:08:19 LOP8M. V8.B. Page 01:01 Codes: 4199 Signs: 2620 . Length: 50 pic 3 pts, 212 mm A SPLITTER FOR THE PETERSEN FAMILY 137 Fig. 1. The seven graphs in the Petersen family. The theory of ``splitters'' was developed by Seymour in [6] (see also [1]). Let F be a family of graphs such that if G # F then any minor of G is also in F. A graph G is called a splitter for F if any 3-connected member of F that contains a G-minor must itself be isomorphic to G. Hence the splitters of a family are the maximal 3-connected graphs that belong to the family. Seymour states that splitters occur ``only in relatively out-of-the-way places'' in graph theory. They are also very useful as possible counterexamples to conjectures about the family in question and can be considered the building blocks of the family. The following theorem is central to the theory of splitters. It was proven by Seymour [6] in 1980 and independently by Tan [7] in 1981. Theorem 1 (Splitter Theorem). Let H be a 3-connected minor of a 3-connected graph G. Further, if H is a wheel, suppose that H is the largest wheel that is a minor of G. Then there exists a chain of 3-connected graphs H=G0m G1m }}}m Gr&1m Gr=G where for 0ir&1, Gi+1 is obtained from Gi by adding an edge or splitting a vertex. Given a 3-connected graph G that is not a wheel, the splitter theorem states that G is a splitter for a family F if G # F and every 3-connected graph obtained from G by adding an edge or splitting a vertex is not in F. Let Q13, 3 be the graph obtained from a circuit of length 13 by adding edges [[ v i , v i +3] |1i13], where the indices are read modulo 13. (See Fig. 2) This graph has been studied previously in several contexts. Thomassen [8] considered Q13, 3 in connection with vertex transitive toroidal embeddings. Randby [2] mentions it because Q13, 3 contains no topological K5 yet it has a 3-representative embedding on the torus. Mohar and the author independently File: 582B 180002 . By:XX . Date:19:01:98 . Time:15:30 LOP8M. V8.B. Page 01:01 Codes: 2395 Signs: 1831 . Length: 45 pic 0 pts, 190 mm 138 JOHN MAHARRY Fig. 2. The graph Q13, 3 . found that the toroidal embedding found by Randby is a minimal 3-repre- sentative toroidal embedding as a corollary to a theorem of Schrijver [5]. A 4-connected graph with no triangles is said to be Tutte-4-connected.A graph which has a vertex, the deletion of which results in a planar graph, is called an apex graph. Robertson [4] had conjectured that there was a unique Tutte-4-connected graph that was not apex and had no member of the Petersen family as a minor, namely the graph obtained by deleting a perfect matching from K5, 5. It turns out that Q13, 3 is a second graph of this type. Moreover, Q13, 3 contains (K5, 5 minus a perfect matching) as a minor. Theorem 2. The graph Q13, 3 is a splitter for the family of graphs with no Petersen family minor. Proof. It is straightforward to verify that Q13, 3 contains no minor isomorphic to any of the Petersen family graphs. By the splitter theorem, all that remains is to check that every 3-connected graph obtained from Q13, 3 by adding an edge or splitting a vertex contains some graph in the Petersen family as a minor. By symmetry, there are only four nonisomorphic graphs obtained by adding an edge to Q13, 3 . These are obtained by adding the edges [v1 , v3], [v1, v5], [v1, v6], and [v1, v7]. Similarly, there are only (up to isomorphism) three minimal ways to split a vertex of Q13, 3 and remain 3-connected. It turns out that each of the seven graphs which are obtained by adding an edge to or splitting a vertex of Q13, 3 contains as minors all seven of the graphs in the Petersen family. This can easily be checked by anyone with sufficient patience. For example, Fig. 3 shows how (K3, 3+v) is a minor of (Q13, 3 plus the edge [v1, v3]) and how the Petersen graph is a minor of (Q13, 3 with a single vertex split). In the figure on the left, the bold edges form a subdivided K3, 3 subgraph and contracting the other vertices to a single vertex results in a K3, 3+v graph. The bold edges in the figure on the right form a subdivided Petersen graph. The other 47 minor inclusions can File: 582B 180003 . By:XX . Date:19:01:98 . Time:15:30 LOP8M. V8.B. Page 01:01 Codes: 2636 Signs: 2055 . Length: 45 pic 0 pts, 190 mm A SPLITTER FOR THE PETERSEN FAMILY 139 Fig. 3. (K3, 3+v)m(Q13, 3+e) and Petersen m(Q13, 3 with a vertex split). be shown similarly. Thus, by the splitter theorem, any 3-connected graph that properly contains Q13, 3 as a minor will also contain every graph in the Petersen Family as a minor. K Corollary 1. Let H be any graph in the Petersen family. The graph Q13, 3 is a splitter for the family of graphs that do not contain a minor isomorphic to H. REFERENCES 1. J. Oxley, ``Matroid Theory,'' Oxford Univ. Press, OxfordÂNew York, 1992. 2. S. Randby, ``Embedding K5 in 4-Connected Graphs,'' Ph.D. Thesis, The Ohio State University, 1991. 3. N. Robertson, P. D. Seymour, and R. Thomas, ``Excluded Minors in Cubic Graphs,'' manuscript. 4. N. Robertson, personal communication,. 5. A. Schrijver, Classification of minimal graphs of given face-width on the torus, J. Combin. Theory Ser. B 61 (1994), 217236. 6. P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305359. 7. J. J-M. Tan, ``Matroid 3-Connectivity,'' Ph.D. thesis, Carleton University, 1981. 8. C. Thomassen, Tilings of the torus and Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991), 605635. File: 582B 180004 . By:XX . Date:28:01:98 .
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